This paper studies the design and application of a novel visual attention model designed to compute user's gaze position automatically, i.e., without using a gaze-tracking system. The model we propose is specifically designed for real-time first-person exploration of 3D virtual environments. It is the first model adapted to this context which can compute in real time a continuous gaze point position instead of a set of 3D objects potentially observed by the user. To do so, contrary to previous models which use a mesh-based representation of visual objects, we introduce a representation based on surface-elements. Our model also simulates visual reflexes and the cognitive processes which take place in the brain such as the gaze behavior associated to first-person navigation in the virtual environment. Our visual attention model combines both bottom-up and top-down components to compute a continuous gaze point position on screen that hopefully matches the user's one. We conducted an experiment to study and compare the performance of our method with a state-of-the-art approach. Our results are found significantly better with sometimes more than 100 percent of accuracy gained. This suggests that computing a gaze point in a 3D virtual environment in real time is possible and is a valid approach, compared to object-based approaches. Finally, we expose different applications of our model when exploring virtual environments. We present different algorithms which can improve or adapt the visual feedback of virtual environments based on gaze information. We first propose a level-of-detail approach that heavily relies on multiple-texture sampling. We show that it is possible to use the gaze information of our visual attention model to increase visual quality where the user is looking, while maintaining a high-refresh rate. Second, we introduce the use of the visual attention model in three visual effects inspired by the human visual system namely: depth-of-field blur, camera motions, and dynamic luminance. All these effects are computed based on the simulated gaze of the user, and are meant to improve user's sensations in future virtual reality applications.